This work is exploring the rectangle as a pattern construction. It is the most recognised geometric shape, can it still provide us with new expressions in fashion? This project is conducted through clear restrictions in the method, and through draping translated into garments through flat pattern construction. The result is a collection with a complex expression, mixing poetic shapes with playful prints full of contrast, which signifes harmony but does not follow the classical notions of beauty. The value of this work lies in the finding of new expressions in fashion, proposing that it is of utmost importance to challenge what we think we know to be true.

The thesis represents a collection of solved problems concerned with covering planar shapes (mostly rectangles with integer sides) by tiles known as polyominoes (e.g., domi- noes, trominoes, tetrominoes, etc.). In most cases, the goal is to find a tiling or to prove that no such tiling exists. In more difficult problems, the task is to deduce conditions for the rectangle to be tileable by specified polyominoes. The last chapter is devoted to calcu- lating the number of all possible tilings of the specified rectangle.

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This thesis studies the open problem of packing rectangles. Is it possible to pack rectangles with dimensions 1/n x 1/(n+1) into a unit square? The aim of this thesis is analysis of the problem and the related algorithm. Attention will be focused mainly on the implementation of this algorithm and on study of its functioning. Powered by TCPDF (www.tcpdf.org)

An report about investigating the rectangular shape in the relation between the stiff and the soft in drapings. The investigation have been made in the field fashion and garments. / Program: Modedesignutbildningen

Emergence is a phenomenon where we study the implicit or hidden meaning of an image. We introduce this concept in image database access and retrieval of images using his as an index for retrieval. This would give an entirely different search outcome than ordinary search where emergence is not considered, as consideration of hidden meanings could change the index of search. A feature of an image, which is not explicit would be emergent feature if it can be made explicit. There are three types of emergence: computational emergence, thermodynamic emergence and emergence relative to a model. In computational emergence, it is assumed computational interactions can generate different features or behaviors. This is one of the approaches in the field of artificial life. Thermodynamic emergence is of the view that new stable features or behaviors can arise from equilibrium through the use of thermodynamic theory. In emergence relative to a model, deviation of the behavior from the original model gives rise to emergence. We would use this latter view in our work. Two classes of shape emergence have been identified: embedded shape emergence and illusory shape emergence. In embedded shape emergence all the emergent shapes can be identified by set theory procedures on the original shape under consideration. For example, in a set S= {a,b,c,d,e}, we can find subsets like S1={a,b,c}, S2={c,d,e}, S3={a,c,e} and so on. But in illusory shape emergence, where contours defining a shape are perceived even though no contours are physically present, this kind of set theory procedures are not enough and more effective procedures have to be applied to find these hidden shapes. These procedures could be based on geometrical, topological or dimensional studies of the original shape. Content-based Image Retrieval (CBIR) techniques, so far developed, concentrated on only explicit meanings of an image. But more meanings could be extracted when we consider the implicit meanings of the same image. To find out the implicit meanings, we first destroy the shape of the original image which gives rise to unstructured image. Then we process the unstructured image to bring out the new emergent image. We discuss emergence, calculation of emergence index and accessing multimedia databases using emergence index in this dissertation. To calculate emergence index in the access of multimedia databases, we take an input image and study the emergence phenomenon of it. Also we study the emergence phenomenon of the images of the database. Both input image and images of database would give rise to more meanings because of emergence as we explained earlier. Based on the new meanings, wherever there would be a match between input image and images of database, we would pick that record up for selection. We defined emergence index as EI = f(D,F,V,C,E) where D stands for domain of the database, F for features of the image, V for various variables that define the image, C for constraints which represent the image and E for emergence phenomenon. We calculate these five variables to get emergence index for each image of the database. Also we calculate these five variables for input image as well. We talk about global aspects of features. It means features of the entire image. Examples are area, perimeter or rectangles, triangles. In some searches, to consider the global features could be advantageous in that a symmetry with the input image could be obtained on the basis of global features only. But as is clearly the case, to consider global features could overlook the individual objects that constitute the image as a whole. In the kind of searches we propose, we take into account the global features of the image of the database while considering in detail local features. Various objects that lie within an image constitute local features. In our example, there are three objects in the image, namely, a lake and two houses. Studying the features of these three objects would add to studying the features of the image globally. We took the example of a geographic location in the thesis and then showed how destruction of original image is done and further processing of the unstructured image gives new emergent image. Partial implementation of this concept is also presented at the end. In implementation, we consider the retrieval of image globally. We do not consider break-up of image into multiple objects which is left for future research.

In this thesis, we introduce a new class of problems, which we call Optimal Areal Positioning (OAP), and study a special form of these problems. OAPs have important applications in earth observation satellite management, tele-robotics, multi-camera control, and surveillance. In OAP, we would like to find the optimal position of a set of floating geometric objects (targets) on a two-dimensional plane to (partially) cover another set of fixed geometric objects (requests) in order to maximize the total reward obtained from covered parts of requests. In this thesis, we consider the special form of OAP in which targets and requests are parallel axes rectangles and targets are of equal size. A predetermined reward is associated with covering an area unit of each request. Based on the number of target rectangles, we classify rectangular OAP into two categories: Single Target Problem (STP) and Multi-Target Problem (MTP). The structure of MTP can be compared to the planar p-center which is NP-complete, if p is part of the input. In fact, we conjecture that MTP is NP-complete. The existing literature does not contain any work on MTP. The research contributions of this thesis are as follows: We develop new theoretical properties for the solution of STP and devised a new solution approach for it. This approach is based on a novel branch-and-bound (BB) algorithm devised over a reduced solution space. Branching is done using a clustering scheme. Our computational results show that in many cases our approach significantly outperforms the existing Plateau Vertex Traversal and brute force algorithms, especially for problems with many requests appearing in clusters over a large region. We perform a theoretical study of MTP for the first time and prove several theoretical properties for its solution. We have introduced a reduced solution space using these properties. We present the first exact algorithm to solve MTP. This algorithm has a branch-and-bound framework. The reduced solution space calls for a novel branching strategy for MTP. The algorithm has a main branch-and-bound tree with a special structure along with two trees (one for each axis) to store the information required for branching in the main tree in an efficient format. Branching is done using a clustering scheme. We perform computational experiments to evaluate the performance of our algorithm. Our algorithm solves relatively large instances of MTP in a short time.

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Piercing problems arise often in facility location, which is a well-studied area of computational geometry. The general form of the piercing problem discussed in this dissertation asks for the minimum number of facilities for a set of given rectangular demand regions such that each region has at least one facility located within it. It has been shown that even if all regions are uniform sized squares, the problem is NP-hard. Therefore we concentrate on approximation algorithms for the problem. As the known approximation ratio for arbitrarily sized rectangles is poor, we restrict our effort to designing approximation algorithms for unit-height rectangles. Our e-approximation scheme requires <I>n</I>^{<I>O</I>(1/&epsilon;??)} time. We also consider the problem with restrictions like bounding the depth of a point and the width of the rectangles. The approximation schemes for these two cases take <I>n</I>^{<I>O</I>(1/&epsilon;)} time. We also show how to maintain a factor 2 approximation of the piercing set in <I>O</I>(log <I>n</I>) amortized time in an insertion-only scenario.

Two players take alternating turns filling a rectangular board with unit squares without rotation, but may be otherwise arbitrary. Squares may not overlap and the game ends when there is no space for the next one. The result of the game is the number of turns. The constructor aims to maximize this quantity while the destructor wants to minimize it. We would like to get close to this value, provided that both players use their optimal strategy. We prove some new lower and upper bound for the game. This thesis extends results given by Tamás Hubai in his paper Competitive rectangle filling. Furthermore, we have a look at other board shapes and shapes to fill with.